Simplicity in nitary abstract elementary
classes
Meeri Kesälä
Kurt Gödel Research Center, University of Vienna
University of Helsinki
Joint work with Tapani Hyttinen
Around classication theory
Workshop at the School of Mathematics,
University of Leeds
27th-30th June, 2008
Question: Can we build an analogue of the F.O. independence calculus in (some subclass of) Abstract Elementary
Classes?
Hence, in the title, simplicity refers to the existence of a wellbehaved notion of independence, which works over sets.
To start with, we might also assume stability, and even that will
not be enough.
1
Independence in non-elementary classes
•
Stable homogeneous classes admit a notion of independence over strongly κ(M)-saturated models, and with simplicity also over sets.
• (ω -stable) excellent classes admit a notion of independence
over (ω-saturated) models, and with simplicity also over sets.
For these classes there is a categoricity transfer theorem and a
Main Gap theorem.
Authors: Keisler, Shelah, Grossberg, Hart, Hyttinen, Lessmann
etc
2
Also:
Hyttinen, Lessmann, Shelah: Interpreting groups and elds in
some nonelementary classes, Journal of Mathematical Logic,
2005.
Conclusion: A lot can be done without compactness.
3
A few words about homogeneous and excellent classes
•
There is no compactness
•
Both motivated by the study of sentences in Lω1ω (or Lκ+ω ),
but we only restrict to sentences where the class of structures
has `good' properties
•
We study F.O. types, but not all types are allowed. There is
a monster model.
•
The two contexts are incomparable, for example, in the h.c.
we assume more amalgamation and in e.c. more stability
4
We study an abstract elementary class (K, 4K) with AP, JEP
and arbitrarily large models.
There K is a class of structures and 4K is `elementary substructure' -relation with certain properties.
(closed under chains, downward Löwenheim-Skolem etc)
For
A, B ∈ K
a K-embedding is an embedding
f :A→B
with
f (A) 4K B.
6
In this context we have a κ-model-homogeneous monster model,
that is,
•
For any
•
For
A ∈ K, |A| < κ,
A 4K M, |A| < κ,
there is a K-embedding
any
f : A → M.
K -embedding
f :A→M
extends to an automorphism of M.
Denition: We say that a set
We say that
A
A⊂M
is bounded, if
is a model if it's bounded,
A∈K
|A| < κ.
and
A 4K M.
7
We also assume countable Löwenheim-Skolem number, that is,
•
For any subset
A⊂M
there is
A 4K M
with
|A| ≤ |A| + ℵ0.
8
Galois type:
tpg (ā/A) = tpg (b̄/A)
there is
f ∈ Aut(M/A)
i
such that
f (ā) = b̄.
Note: Types over sets make only sense for subsets of the mon-
ster model, since we don't have amalgamation for sets.
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We also dene a notion of type with nite character, so called
weak type:
tpw (ā/A) = tpw (b̄/A)
tpg (ā/A0) = tpg (b̄/A0)
i
for all nite
A0 ⊂ A.
Note: In excellent and homogeneous classes these two notions
agree over models.
10
We say that a sequence
(āi)i<λ
is strongly A-indiscernible if
•
it can be extended to
•
Any order-preserving partial f
mapping each āi to āf (i).
(āi)i<λ0
Lemma 1 For any bounded
for any bounded
: λ → λ extends to F ∈ Aut(M/A)
A⊂M
|X| ≥ i
and any nite
(a0, ..., an) ⊂ X .
and
2(|A|+LS(K))
n
λ0 > λ.
there is a strongly
X⊂M
s.t.
+
A-indiscernible (ai)i<ω
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s.t.
We say that tpw (ā/A) Lascar splits over nite E ⊂ A if there is
a strongly E -indiscernible sequence (āi)i<ω such that ā0, ā1 ∈ A
and
tpg (ā0/E ∪ ā) 6= tpg (ā1/E ∪ ā).
We write that
ā ↓A B
if there is nite E ⊂ A such that for each D ⊇ B there is b̄
realizing tpw (ā/A ∪ B) such that tpw (b̄/D) does not Lascar split
over E .
12
We say that (K, 4K) is simple if for each tuple
A we have that
ā
and nite set
ā ↓A A
.
Note that requiring this for models
A
is a dierent thing.
13
We dene that (K, 4K) is (weakly) λ-stable, if the number of
weak types over a set of size ≤ λ is ≤ λ.
We dene that that (K, 4K) is superstable if it is weakly λ-stable
for some λ and there are no ā and nite An, n < ω such that
S
• n<ω An
is a model
• ā ↓An An+1
for all
n < ω.
14
A practise Lemma: Assume that
λ-stable
for some λ. Then
↓
(K, 4K)
satises
is simple and weakly
•
invariance, monotonicity, extension (by denition)
•
symmetry over nite sets
•
transitivity
•
stationarity for Lascar strong types
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We "almost" get results such that
•
simplicity and ω-stability imply superstability
•
simplicity and superstability imply all the usual properties of
independence, stationarity for Lascar strong types
• ω -stability,
plicity
•
`tameness' for types and nite U -rank imply sim-
simplicity and `tameness' for types imply a categoricity transfer theorem
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What "almost" means: We also need to assume
1. Finite character of 4K.
2. Tarski-Vaught property for 4K
(This also follows from 1. and weak ω-stability.)
Note: These hold if
Lω 1 ω ,
4K is given by a countable fragment of
these hold in homogeneous and excellent classes
17
Questions: Are these the `right' generalizations of such no-
tions? Are there other notions, are there many dierent notions
for dierent `applications' ?
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But: There is a
class which is not simple, although it is homogeneous, excellent and totally categorical (hence
also ω-stable)
Lω1ω -denable
Also: If (K, 4K) would admit any notion of independence which
would satisfy
•
invariance, monotonicity, extension, transitivity
•
local character (for any ā,B there is nite
•
bounded number of free extensions of a type (over a nite
set)
E⊂B
s.t.
ā ↓E B )
then it would be simple.
19